Nyquist filter and method

ABSTRACT

An improved Nyquist filter can be used as a matched filter in a digital communications system. This filter is characterized in that the frequency domain response meets the Nyquist criteria and that the square root of the frequency domain response has a first derivative that is continuous at all points. In other embodiments, the square root of the frequency domain response continuous over all points for all higher order derivatives.

FIELD OF THE INVENTION

[0001] The present invention relates generally to filter circuits andmore particularly to an improved Nyquist filter and method.

BACKGROUND OF THE INVENTION

[0002] The present invention relates to filters such as the kind thatcan be utilized in communications systems. FIG. 1 illustrates a basicblock diagram of a digital communication system 10 that utilizes pulseamplitude modulation. In this system, a pulse generator 12 receivesclock pulses and binary input data. The output of pulse generator 12will be a digital binary stream of pulses.

[0003] The pulse stream from pulse generator 12 is applied to thedigital transmitting filter 14 that shapes the pulse for output to thedigital-to-analog converter 15 and transmission over channel 16. Channel16 may be a wired or wireless channel depending upon the application.The transmitted data is received at receiving filter 18. The output offilter 18 is applied to analog-to-digital converter 20.Analog-to-digital converter 20 utilizes clock pulses that are generallyrecovered from the transmitted data by clock recovery circuit 22. Theoutput binary data from analog-to-digital converter 20 is a replica ofthe input binary stream that was provided to pulse generator 12.

[0004] Major objectives of the design of the baseband PAM system are tochoose the transmitting and receiving filters 14 and 18 to minimize theeffects of noise, to eliminate or minimize inter-symbol interference(ISI) and to reduce stop band energy. Inter-symbol interference can betheoretically eliminated by properly shaping the pulses of thetransmitted signal. This pulse shaping can be accomplished by causingthe pulse to have a zero value at periodic intervals.

[0005] Modern embodiments of pulse shaping filters use a pair of matchedfilters, one for transmit and one for receive. The convolution of thetransmit filter with the receive filter forms the complete pulse shapingfilter. Inter-symbol interference is avoided since the combined filterimpulse response reaches unity at a single point and is zeroperiodically at every other information point (Nyquist sampling rate).The linear superposition of pulses representing a pulse train preservesbandwidth and information content. Linear superposition of band limitedpulses remains band limited and sampling the combined filter at theinformation rate recovers the information.

[0006]FIG. 3b shows an example of a Nyquist filter impulse response.Zeros occur at the information rate, except at one information bearingpoint. All Nyquist filters having the same stop band are equallybandwidth limited if the time response of the filters is allowed to goto infinity. Realizable filters, however, are truncated in time since itis not possible to have an infinitely long time function. Truncationerror in the time domain causes the theoretical stop band achievable byall Nyquist filters to be violated, so that out of band energy exists inexcess of the stop band frequency.

[0007] The most bandwidth efficient filter is the “brick wall” filterillustrated in FIG. 3a by the box (α=0). The time response of thisfilter is shown in FIG. 3b (α=0). While bandwidth efficiency istheoretically greatest for a brick wall filter as the time responseapproaches infinity, truncation error causes poor performance forpractical and realizable approximations to the brick wall filter.

[0008] One method of producing practical filters is to allow the stopband of Nyquist compliant filters to exceed the bandwidth of the idealbrick wall filter and smoothly transition to the stop band. A class ofsuch filters is the raised cosine filters. In the frequency domain (FIG.3a), the raised cosine filter smoothly approaches the frequency stopband (except for the limiting brick wall filter case). The raised cosinefilter is continuous at the stop band and the first derivative iscontinuous. The second derivative of a raised cosine filter, however, isnot continuous at the stop band.

[0009] In current embodiments of most systems, the raised cosine filteris used in its matched filter version. The transmit square root raisedcosine filter, which determines the spectral bandwidth efficiency of thesystem, is discontinuous in the first derivative at the stop band.

SUMMARY OF THE INVENTION

[0010] The preferred embodiment of the present invention utilizes apulse shaping filter that meets the Nyquist criteria. This filter alsohas the property of being continuous in the frequency domain up to andincluding the first derivative for the square root matched filterversion. Some embodiments of the invention are in fact continuous in allderivatives for the square root version and these filters are closer tothe ideal brick wall filter for the same stop band.

[0011] Nyquist filters are produced from filters in the frequency domainwith a fixed frequency cutoff. As is well known in the state of the art,a fixed cutoff frequency leads to an unrealizable filter of infiniteduration in the time domain. To produce a realizable filter, the idealfilter is approximated by time delaying and truncating the infiniteimpulse response. Truncation, however, produces unintentional out ofband energy. One goal that is achieved by some embodiments of thepresent invention is to minimize this unintentional out of band energyafter the filter is truncated.

[0012] Embodiments of the present invention provide filters that give asmaller signal ripple at the truncation length for the same theoreticalstop band as raised cosine filters and therefore have better attenuationin the frequency domain. Accordingly, the preferred embodiment of thepresent invention has better truncation performance than the raisedcosine that represents state of the art design for identical theoreticalstop band.

[0013] The present invention includes embodiments that have severaladvantages over prior art Nyquist filters such as the raised cosinefilter. For example, the filter of the preferred embodiment of thepresent invention reduces the effects of truncation errors, by reducingthe energy remaining in the terms beyond the truncation length. Thisattenuation leads to lower energy levels in the stop bands. For example,one embodiment filter of the present invention has been shown to provide10 dB improvement in the filter stop band, as compared with a comparableraised cosine filter. In other words, the out-of-band transmissions arereduced by 90%, a significant improvement.

[0014] Implementation of embodiments of the present invention in acommunication system provides enhanced system performance. Since theout-of-band performance is improved, adjacent channels can be movedcloser together and use less frequency guard band. This feature leads tomore efficient use of the available bandwidth. This advantage will besimilarly be attained for subchannels within a channel.

BRIEF DESCRIPTION OF THE DRAWINGS

[0015] The above features of the present invention will be more clearlyunderstood from consideration of the following descriptions inconnection with accompanying drawings in which:

[0016]FIG. 1 is a block diagram of a known digital communication system;

[0017]FIG. 2 is a block diagram of a QAM communication system that canutilize the teachings of the present invention;

[0018]FIGS. 3a and 3 b are the frequency and impulse responsecharacteristics of a raised cosine filter;

[0019]FIG. 4 is a Nyquist filter shown as being represented by threeregions;

[0020]FIG. 5 is a plot comparing the frequency response of a knownraised cosine filter with a preferred embodiment Nyquist filter of thepresent invention;

[0021]FIGS. 6a and 6 b are an impulse response of a filter of thepresent invention;

[0022]FIGS. 6c and 6 d compare impulse response for two embodimentfilters of the present invention;

[0023]FIG. 7 is a generalized diagram of an RF communications systemthat can utilize the present invention;

[0024]FIGS. 8a-8 c show an exemplary base station of the system of FIG.7; and

[0025]FIGS. 9a-9 b show an exemplary terminal unit of the system of FIG.7.

DETAILED DESCRIPTION

[0026] The making and use of the presently preferred embodiments arediscussed below in detail. However, it should be appreciated that thepresent invention provides many applicable inventive concepts that canbe embodied in a wide variety of specific contexts. The specificembodiments discussed are merely illustrative of specific ways to makeand use the invention, and do not limit the scope of the invention.

[0027] The present invention will first be described with respect to aquadrature amplitude modulation (QAM) system. An improved class offilters will be described with respect to this system. A number of otherapplications that could also utilize a filter of the present inventionare then discussed.

[0028]FIG. 2 illustrates a block diagram of a QAM system 100 that canutilize the filter of the present invention. As shown by blocks 102 and104, the system can be used for either analog or digital data, or both.If an analog signal, such as voice and/or video as examples, is to betransmitted, it will first be filtered with low pass filter (LPF) 106and then converted to a digital signal by analog-to-digital converter(ADC) 108. The digital stream, whether from digital source 104, ADC 108,or both, will then be mapped into I (in-phase) and Q (quadrature-phase)carriers in mapping unit 110. Each of these steps is well known in theart and will not be described in detail herein. For more informationregarding these blocks, reference can be made to W. T. Webb and L.Hanzo, Modern Quadrature Amplitude Modulation, IEEE Press, 1994, chapter3, pp. 80-93, incorporated herein by reference. FIGS. 2 and 3 wereadapted from this text.

[0029] The I and Q streams will be filtered in Nyquist filters 112 and112′. As will be discussed below, the present invention provides aNyquist filter 112 that has enhanced performance compared to prior artfilters. In general, a Nyquist filter has an impulse response withequidistant zero-crossings at symbol points. As a result, the filtereliminates inter-symbol interference (ISI). More detail of the improvedNyquist filter 112 (112′) will be provided below.

[0030] Once the I and Q signals have been generated and filtered, theyare modulated by an I-Q modulator 114. The modulator 114 includes twomixers 116 and 118. As shown, mixer 116 is used for the I channel andmixer 118 is used for the Q channel. The modulator 114 causes both the Iand Q channels to be mixed with an intermediate frequency (IF) signal,generated from signal source 120. The I channel will be mixed with a IFsignal that is in phase with respect to the carrier and the Q channelwill be mixed with an IF signal that is 90 degrees out of phase. Thisprocess allows both signals to be transmitted over a single channelwithin the same bandwidth using quadrature carriers.

[0031] The analog signal output from the modulator 114 is then frequencyshifted to the carrier frequency by modulator 124. In the preferredembodiment, the carrier frequency is in the radio frequency (RF) range,but other frequencies could be used. The present invention would equallyapply to any system that uses CDMA (code division multiple access), TDMA(time division multiple access), optical systems, HDTV (high definitiontelevision), cable systems and others.

[0032] Returning to FIG. 2, the RF signal is transmitted to the receiverthrough a channel 126. This channel can be wireless, e.g., RF wirelesscommunications. Alternatively, the channel could be an electricalconnection or an optical connection.

[0033] A demodulator 128 at the receiver demodulates the received signalby mixing it down to the IF for the I-Q demodulator 130. The I-Qdemodulation takes place in the reverse order to the modulation process.The signal is split into two paths, with each path being mixed down withintermediate frequencies that are 90° apart. The two paths are thenprovided to Nyquist filters 132 and 132′, which can be of the typedescribed below.

[0034] The output of the Nyquist filters 132 and 132′ is provided to ademapping unit 134 which returns the signal to a digital stream. If theoriginal data was digital, then the data from digital source 104 shouldbe recovered at digital output 136. If the original signal was analog,on the other hand, the digital stream from demapping unit 134 would bereturned to analog form by digital-to-analog converter (DAC) 138. Thisanalog output of DAC 138 could then be filtered by low pass filter 140and provided to analog output 142. Once again, reference can be made tothe Webb and Hanzo text for additional details on QAM systems.

[0035] As mentioned above, a Nyquist filter has an impulse response withequidistant zero-crossings at sampling points to eliminate inter-symbolinterference (ISI). FIGS. 3a and 3 b show the frequency characteristic(FIG. 3a) and impulse response (FIG. 3b) of a well known Nyquist filter,the raised cosine filter. Nyquist showed that any odd-symmetricfrequency domain extension characteristic about f_(N) and (−f_(N))yields an impulse response with a unity value at the correct signalinginstant and zero crossings at all other sampling instants. The raisedcosine characteristic meets these criteria by fitting a quarter periodof a frequency-domain cosine shaped curve to an ideal (brick wall)filter characteristic.

[0036] The parameter controlling the bandwidth of the raised cosineNyquist filter is the roll-off factor α. The roll-off factor α is one(α=1) if the ideal low pass filter bandwidth is doubled, that is thestop band goes to zero at twice the bandwidth (2f_(N)) of an ideal brickwall filter at f_(N). If α=0.5 a total bandwidth of 1.5f_(N) wouldresult, and so on. The lower the value of the roll-off factor α, themore compact the spectrum becomes but the longer time it takes for theimpulse response to decay to zero. FIGS. 3a and 3 b illustrate threecases, namely when α=0, α=0.5 and α=1.0.

[0037] The equation defining the raised cosine filter in the frequencydomain (NF_(Raised Cosine)(f) where f is frequency) and thecorresponding impulse response (nf_(Raised Cosine)(t) where t is time)are defined by the following equations. $\begin{matrix}{{{NF}_{RaisedCosine}(f)} = \left\{ \begin{matrix}\begin{matrix}{T,{{{for}\quad 0} \leq {f} \leq \frac{1 - \alpha}{2\quad T}}} \\{{{\frac{T}{2}\left( {1 - {\sin \quad\left\lbrack {\frac{\pi \quad T}{\alpha}\left( {f - \frac{1}{2\quad t}} \right)} \right\rbrack}} \right)},{{{for}\quad \frac{1 - \alpha}{2\quad T}} < {f} \leq \frac{1 + \alpha}{2\quad T}}}\quad}\end{matrix} \\{0,{{for}\quad {all}\quad {other}\quad {values}\quad {of}\quad f}}\end{matrix} \right.} \\{{{nf}_{RaisedCosine}(t)} = {\frac{\sin \left( \frac{\pi \quad t}{T} \right)}{\frac{\pi \quad t}{T}}\frac{\cos \left( \frac{\alpha \quad \pi \quad t}{T} \right)}{\left( {1 - \frac{4\quad \alpha^{2}t^{2}}{T^{2}}} \right)}}}\end{matrix}$

[0038] Matched filters are used in many communication systems in orderto maximize the signal-to-noise ratio. As illustrated in FIG. 2, thematched filtering can be accomplished by including Nyquist filters atboth the transmitter (filter 112) and the receiver (filter 132). Sincetwo filters are provided, each will have a characteristic of square rootof a Nyquist function in the frequency domain. In this manner when theeffect of both filters is taken into consideration, the desired Nyquistcharacteristic will be achieved. In other words, the product in thefrequency domain of the two matched filters is equivalent to thefrequency domain representation of the Nyquist filter. The followingequations provide the frequency domain and impulse response of thesquare root version of the known raised cosine filter. $\begin{matrix}{{{NF}_{{Sqrt}\text{-}{RC}}(f)} = \left\{ \begin{matrix}\begin{matrix}{\sqrt{T},{{{for}\quad 0} \leq {f} \leq \frac{1 - \alpha}{2\quad T}}} \\{{{\sqrt{\frac{T}{2}}\left( {1 - {\sin \quad\left\lbrack {\frac{\pi \quad T}{\alpha}\left( {f - \frac{1}{2\quad t}} \right)} \right\rbrack}} \right)^{1/2}},{{{for}\quad \frac{1 - \alpha}{2\quad T}} < {f} \leq \frac{1 + \alpha}{2\quad T}}}\quad}\end{matrix} \\{0,{{for}\quad {all}\quad {other}\quad {values}\quad {of}\quad f}}\end{matrix} \right.} \\{{{nf}_{{Sqrt}\text{-}{RC}}(t)} = {\frac{4\quad \alpha}{\pi \sqrt{T}}\frac{\left\{ {{\cos \left\lbrack {\left( {1 + \alpha} \right)\frac{\pi \quad t}{T}} \right\rbrack} + \frac{T\quad {\sin \left\lbrack {\left( {1 - \alpha} \right)\pi \quad {t/T}} \right\rbrack}}{4\quad \alpha \quad t}} \right\}}{\left\lbrack {1 - \left( {4\quad \alpha \frac{t}{T}} \right)^{2}} \right\rbrack}}}\end{matrix}$

[0039] Spectral efficiency can be gained using filters that are Nyquistcompliant and that smoothly transition to the stop band in the frequencydomain. FIG. 4 shows three approximate regions of a Nyquist filter inthe frequency domain. Region I constitutes the unattenuated passband.Region II represents the transition band and Region III represents thestop band. Filter smoothness can be measured by the number ofderivatives with respect to frequency that remain continuous. Inparticular, for a good filter performance, the filter should be smoothat the point between the transition and stop band regions (II and II)and at the point between the pass band and transition regions (I andII). The square root raised cosine is discontinuous in the firstderivative at the point between regions II and III.

[0040] The envelop of the time domain response decays more rapidly ifthe frequency response is smooth, that is continuously differentiable.Unfortunately, the frequency response of a raised cosine filter isnon-ideal when the square root is taken. In particular, when the squareroot of a raised cosine filter is taken, the first derivative isdiscontinuous at the boundary of regions II and III. As a result, thefrequency domain curve does not come in smooth at the stopband as in theraised cosine case (see FIG. 3a). Instead, the curve comes in sharply tothe stopband. A discontinuity in the frequency domain can lead to higherpeaking at longer duration in the time domain.

[0041]FIG. 4 can be used to illustrate this concept. As discussed above,a Nyquist filter (either of the prior art or of the present invention)is shown as being divided into three regions. Region I is the pass band,region II is the transition band and region III is the stop band. Thetransition points between regions, labeled 405 and 410, are the pointsof greatest concern since the functional form of the frequency domainresponse is defined by different equations at these points. One goal ofthe preferred embodiment of the present invention is that the filterfunction be not only continuous but also smooth (i.e., continuous for atleast the first derivative) in both the Nyquist filter and square rootNyquist version.

[0042] The present invention provides a class of Nyquist filters thathave better stop band performance than the known raised cosine filter.The Nyquist filter of the preferred embodiment of the present inventioncomes in smooth at the stopband when the square root version is used. Inother words, the first derivative of this function is continuous. Thisproperty allows for truncation at some delay with less energy loss thanthe known raised cosine.

[0043] In the preferred embodiment, an improved class of Nyquist filterswill meet two criteria. First, the filter will meet the Nyquistcriteria, that is the frequency domain will have odd symmetry about thecutoff frequency (f_(N)). Second, the square root version of thefrequency domain of the filter will be continuous at least in the firstderivative, preferably at all points. In a subclass of filters of thepresent invention, all derivatives will be continuous at all points. Intypical applications, the higher order derivatives will all be equal tozero at the transition point between regions II and III and between Iand II.

[0044] A number of filters meet the criteria for the class of filters ofthe preferred embodiment of the invention. Examples include all Nyquistcompliant functions that when differentiated can be written in the form${F^{\prime}(\omega)} = {{f^{\prime}(\omega)}\cos \left\{ {\frac{\pi}{2}{\sin \left( {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\} {\cos \left\lbrack {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}}$

[0045] with stop band π/T(1+α). Here derivatives to all orders aresmooth at the stop band giving zero (using the product rule fordifferentiation). As will be discussed in further detail below,preferred examples of such functions are composite sine and cosinefunctions. Other functional forms may also meet this criteria.

[0046] The first example of an improved Nyquist filter has a frequencydomain defined by a composite sine function. In particular, thefrequency characteristic of this preferred filter is provided by thefollowing equations: $\begin{matrix}{{{NF} = T},{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}} \\{{{{NF} = {\frac{T}{2}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}} \right\}}} \right)}},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} \leq {\omega } \leq {\frac{\pi}{T}\left( {1 + \alpha} \right)}}}\quad} \\{{{NF} = 0},{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\end{matrix}$

[0047] The square root of the frequency domain is given by the followingequations. $\begin{matrix}{{\sqrt{NF} = \sqrt{T}},{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}} \\{{{\sqrt{NF} = {\sqrt{\frac{T}{2}}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}} \right\}}} \right)^{1/2}}},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} \leq {\omega } \leq {\frac{\pi}{T}\left( {1 + \alpha} \right)}}}\quad} \\{{\sqrt{NF} = 0},{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\end{matrix}$

[0048] This equation was derived by starting with the raised cosinefrequency domain function and taking π/2 times the sine term of thatraised cosine function as the argument of a second sine function.Accordingly, the frequency characteristic is a composite function sinceit includes a sine of a sine. This modification was made because whenthe square root version is differentiated, the frequency domain functionis smooth at the stop bands. This function is continuous in thefrequency domain up to and including the first derivative since thefunction and the first derivative of the function are continuous.

[0049] The time response can be derived by inverting the frequencyfunction. This is done by taking the theoretical Fourier transform ofthe frequency function and leads to the following response:${n\quad {f(t)}} = {\frac{1}{\pi}\left( {{\pi \quad {{sinc}\left( \frac{t}{T} \right)}{\cos \left( \frac{\pi \quad \alpha \quad t}{T} \right)}} + {2\left( {2\quad \alpha} \right)^{2}\left( \frac{t}{T} \right){\sin \left( \frac{\pi \quad t}{T} \right)}{\cos \left( \frac{\pi \quad \alpha \quad t}{T} \right)}{\sum\limits_{m = 1}^{\infty}\quad {\left( {- 1} \right)^{m + 1}{{J_{{2m} - 1}\left( \frac{\pi}{2} \right)}\left\lbrack \frac{1}{\left( {{2m} - 1} \right)^{2} - \left( \frac{2\quad \alpha \quad t}{T} \right)^{2}} \right\rbrack}}}}} \right)}$

[0050] The impulse response of the square root Nyquist filter can beexpressed as $\begin{matrix}{{{nf} - {{sqrt}(t)}} = {{\frac{1 - \alpha}{\sqrt{T}}\left( {1 - {\frac{\sqrt{2}}{2}{J_{0}\left( \frac{\pi}{4} \right)}}} \right){{sinc}\left( {\frac{t}{T}\left( {1 - \alpha} \right)} \right)}} + {\frac{\sqrt{2}}{2\sqrt{T}}{J_{0}\left( \frac{\pi}{4} \right)}\left( {1 + \alpha} \right){{sinc}\left( {\frac{t}{T}\left( {1 + \alpha} \right)} \right)}} +}} \\{{{\frac{\left( {2\alpha} \right)^{2}2\sqrt{2}}{\pi \sqrt{T}}\left( \frac{t}{T} \right){\sin \left( \frac{\pi \quad t}{T} \right)}{\cos \left( \frac{\pi \quad \alpha \quad t}{T} \right)}{\sum\limits_{m = 1}^{\infty}{\left( {- 1} \right)^{m + 1}\frac{J_{{2m} - 1}\left( \frac{\pi}{4} \right)}{\left\lbrack {\left( {{2m} - 1} \right)^{2} - \left( \frac{2\quad \alpha \quad t}{T} \right)^{2}} \right\rbrack}}}} +}} \\{{\frac{\alpha^{2}2\sqrt{2}}{\sqrt{T}\pi}\left( \frac{t}{T} \right){\cos \left( \frac{\pi \quad t}{T} \right)}{\sin \left( \frac{\pi \quad \alpha \quad t}{T} \right)}{\sum\limits_{m = 1}^{\infty}{\left( {- 1} \right)^{m + 1}\frac{J_{2m}\left( \frac{\pi}{4} \right)}{\left\lbrack {m^{2} - \left( \frac{\quad {\alpha \quad t}}{T} \right)^{2}} \right\rbrack}}}}}\end{matrix}$

[0051] It is noted that mathematically inverting the frequency domainfunction leads to an infinite number of terms. This is not a problem,however, since the higher order terms decrease rapidly. In effect, thewaveform can be calculated using only the first few terms.

[0052] As can be seen from the above equation, the impulse responseincludes Bessel functions J(x). In a digital computing system, thesefunctions can be derived in the same manner as any other function, suchas a sine. As a result, the present invention is no more difficult toimplement than any other filter.

[0053] In the preferred embodiment, the Nyquist filter is implemented asa digital filter. Accordingly, the impulse response equation can becalculated once and the results stored in a lookup table. As a result,the fact that the equation is computationally complex is not adetriment. Since the equation is solved off-line, there is littlenegative impact if the computation is time consuming. Of course, thisfact does not prevent the impulse response from being calculated inreal-time if a system was so designed.

[0054] The performance characteristic of the first embodiment filter isillustrated in FIG. 5 along with curves from a square root raised cosinefilter where α=0.17 and α=0.2. Each of these curves are taken in thesquare root version. These curves were taken from simulations of asystem as described in co-pending application Ser. No. ______ (COM-002).In that system, a roll-off factor α=0.17 was found to be optimal for araised cosine filter. As demonstrated by FIG. 5, the performance of thenew Nyquist filter is improved at the stopbands, i.e., outside the idealbandwidth-of the channel. As shown in the figure, the filter of thepresent invention has 10 dB lower transmissions at the stop band whentruncated to eight symbol periods. This represents a significantimprovement.

[0055] Improved stop band performance is beneficial in a communicationsystem since it allows signals on adjacent frequency channels to be morecompactly fit into the frequency spectrum. It also makes it easier tomeet the emissions mask requirements for a given channel. These maskrequirements are generally dictated by a regulatory authority, such asthe Federal Communications Commission in the United States, and definethe level of signal that can be allowed outside a given channel.

[0056] In addition this filter eventually has lower amplitude ringingill the tails than a raised cosine filter. The new class of filters arecharacterized in the time domain by the tails damping faster than theequivalent raised cosine filter tails. After a given time delay in theimpulse response, the new Nyquist filters ring at a lower amplitude andcontinue to ring at a lower amplitude than the equivalent raised cosinefilter at all longer times to infinity.

[0057]FIGS. 6a and 6 b illustrate an exemplary impulse response for thenew Nyquist filter (curve 610) and a known raised cosine filter.Referring to FIG. 6b in particular, the new Nyquist filter exhibits muchlower amplitude levels at times farther from the peak. Because of theselower amplitudes, less energy is lost by truncation and thereforetruncation will have a smaller affect on the frequency domain (and thusthe stop band improvement shown in FIG. 5). In the filter of thepreferred embodiment, the time domain is truncated after eightsymbol-time delays, e.g., after 8T where T is the symbol rate.

[0058] A second embodiment filter of the present invention will now bedescribed. This filter exhibits each of the characteristics of the firstembodiment filter. That is, the filter meets the Nyquist criteria and,in the square root version, is continuous for all values of the firstderivative. In addition, this filter is also continuous for all valuesof higher order derivatives as well.

[0059] The frequency domain and square root frequency domain equationsfor the second embodiment filter can be expressed with the followingequations: $\begin{matrix}{{N(\omega)} = \left\{ \begin{matrix}\begin{matrix}{{T,{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}}\quad} \\{{\frac{T}{2}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{\pi}{2\quad}{\sin \left( {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\rbrack}} \right\}}} \right)},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} < {\omega } < {\frac{\pi}{T}\left( {1 + \alpha} \right)}}}\end{matrix} \\{{0,{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\quad}\end{matrix} \right.} \\{{N_{Sqrt}(\omega)} = \left\{ \begin{matrix}\begin{matrix}{{\sqrt{T},{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}}\quad} \\{{\sqrt{\frac{T}{2}}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{\pi}{2\quad}{\sin \left( {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\rbrack}} \right\}}} \right)^{1/2}},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} < {\omega } < {\frac{\pi}{T}\left( {1 + \alpha} \right)}}}\end{matrix} \\{{0,{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\quad}\end{matrix} \right.}\end{matrix}$

[0060] Inversions of the square root filters for the first and secondembodiments are shown in FIG. 6c (and FIG. 6d, which focuses on thetails). The second embodiment inversion was performed numerically. Bothfilters have the same theoretical stop band. The second embodimentexhibits the same behavior with respect to the first embodiment as thefirst embodiment does with respect to the raised cosine filter. As shownin the figure, the first embodiment rings less at the beginning than thesecond embodiment. As shown in FIG. 6d, however, the second embodimentdamps out more quickly giving less energy in the tails. The secondembodiment transitions more smoothly to the stop band than the firstembodiment.

[0061] Two examples of filters that meet the criteria for the new classof Nyquist filters have been described. Other functional forms may alsomeet the criteria of continuous derivatives and can be considered forthis type of filters. These expressions might include hyperbolic sinesand cosines, polynomials, and ecliptic functions.

[0062] Alternatives to the preferred embodiments also include linearcombinations of terms in the transition band. For example, a filtercould be derived by adding the frequency domain response of the firstembodiment described above with the frequency domain response of thesecond embodiment described above. These two (or more) functions couldbe weighted evenly or not. The terms could each include a differingnumber of sine terms in the combinations. Filters with transitionalregion (region II in FIG. 4) terms like$\frac{T}{2}\left( {1 - {\frac{1}{\sin (a)}\left( {\sin \left\lbrack {a\quad \sin \frac{\pi}{2}{\sin \left( {\frac{T}{{2\alpha}\quad}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\rbrack} \right)}} \right)$

[0063] and square root versions of the same, where a is to bedetermined, and linear combinations of these terms, can be fashioned tomeet Nyquist's criteria.

[0064] Filters as described herein can be used in a variety ofapplications. For example, the filters can be used in any system thatutilizes pulse shaping filters. Digital communication systems provideone such example. For example, filters of the present invention could beused in wireless communications (cellular, GSM, microwave, satellite),wired communications (in telephone systems, cable modems), opticalsystems, broadcast systems (digital television/radio, satellite), andothers.

[0065] One example of a system that can utilize a filter of the presentinvention is described in co-pending application Ser. No. ______(COM-002), which is incorporated herein by reference as if reproduced inits entirety. A Nyquist filter of the present invention can be used inplace of each Nyquist filter included in that system. This system willnow be described with respect to FIGS. 7-9.

[0066]FIG. 7 illustrates an exemplary radio system 700. System 700 couldbe a cellular telephone system, a two-way radio dispatch system, alocalized wireless telephone or radio system or the like. Base unit 702can communicate over transmission medium 704 to one or more terminalunits 706. Transmission medium 704 represents the wireless communicationspectrum. Terminal units 706 can be mobile units, portable units, orfixed location units and can be one way or two way devices. Althoughonly one base unit is illustrated the radio system 700 may have two ormore base units, as well as interconnections to other communicationsystems, such as the public switched telephone network, an internet, andthe like. In the preferred embodiment, the system provides for fullduplex communications. The teachings of the present invention, however,apply equally to half duplex systems, simplex systems as well as to timedivision duplex and other two-way radio systems.

[0067] Details of preferred embodiments of the base unit and terminalunits are provided in the following descriptions. FIGS. 8a through 8 cprovide block diagrams of several major components of an exemplary baseunit 702. A skilled practitioner will note that several components of atypical radio transmitter/receiver not necessary to an understanding ofthe invention have been omitted. Note that many of the features andfunctions discussed below can be implemented in software running on adigital signal processor or microprocessor, or preferably a combinationof the two.

[0068]FIG. 8a illustrates the four sub-channel architecture of a baseunit 102 operating in transmitter mode. The following explanation willbe addressed to sub-channel A, although the teachings apply to the othersub-channels as well. Sub-channel A includes “red” signal coding block802 and “blue” signal coding block 804. The “red” and “blue” arearbitrary designations for the first and second time slots. Detailsregarding the signal coding blocks are provided in the co-pendingapplication. For the present purpose, it is sufficient to state that thesignal coding blocks receive voice and/or data signals, encode thosesignals if necessary, combine control signals, and prepare the combinedsignals for passage to QAM modulator 806.

[0069] Modulator 806 modulates the received signal using a quadratureamplitude modulation (QAM) architecture employing a 16 pointconstellation. With a sixteen point constellation, each symbol mapped tothe constellation represents four bits. In the preferred embodiments,the signal is differentially encoded using a differential Gray codealgorithm. Details of such architectures are well known in the art. See,for instance, Webb et al., Modern Quadrature Amplitude Modulation (IEEEPress 1994). Various other QAM techniques are known in the art,including Star QAM, Square QAM, and Coherent QAM. Additionally, otherencoding techniques could be used, such as Okunev encoding orKhvorostenko encoding could be used in lieu of Gray coding. Otherembodiments of the invention could include other modulation techniquesas are known in the art, provided the modulation provides for sufficientdata rate (16 kb/s in the preferred embodiments) with acceptable signalquality (i.e. signal to noise ratio) for the desired application.

[0070] The in-phase and quadrature-phase components of the QAM modulatedsignal (illustrated by the single line representing both signals, asindicated by the slash through each such signal path) are then passed toNyquist filter 808 which provides a pulse shaping filter in order tolimit the overall bandwidth of the transmitted signal. In the preferredembodiment, the Nyquist filter operates at a 65 times over-samplingrate, in order to simplify the analog filtering of digital images.Nyquist filter 808 may comprise any of the filters discussed in thisspecification.

[0071] Additionally, the signal is multiplied in sub-channel offsetblock 810 by the offset required for the sub-channel upon which thesignal is to be transmitted. In the preferred embodiment, the offset forsub-channel A would be minus 7.2 kHz, for sub-channel B would be minus2.4 kHz, for sub-channel C would be plus 2.4 kHz, and for sub-channel Dwould be plus 7.2 kHz.

[0072] The QAM modulated and filtered signal A is then combined with themodulated and filtered signals from sub-channels B, C, and D insub-channel summer 812 before being passed to digital-to-analogconverter 814 where the combined signals are converted to an analogsignal. The signal is then passed to radio frequency circuitry (notshown) where the signal is modulated to RF and amplified fortransmission, as is known in the art.

[0073] In the preferred embodiment, the system is implemented using adigital signal processor. In this embodiment, all of the circuitry inthe box labeled “Subchannel A” (as well as the other subchannels) andthe subchannel summer 812 are implemented by a single chip. In fact,this chip can be designed to handle analog signals at both the input andthe output by integrating an analog-to-digital converter (not shown) anddigital-to-analog converter 814 on chip. Alternatively, the functionscan be distributed amongst a number of interconnected integrated circuitchips.

[0074] In the preferred embodiment, the impulse response of filter 808would be calculated once and stored in a lookup table, for example aEEPROM on a digital signal processor chip. The filter can then beimplemented in the same manner as any other finite impulse response(FIR) filter, as is known in the art.

[0075]FIGS. 8b and 8 c illustrate base unit 102 operating in receivermode. FIG. 8b provides a high level block diagram of the foursub-channel architecture. Signals from the terminal units are receivedby radio frequency (RF) receiving circuitry (not shown). A/D converter840 receives the signal from the RF receiving circuitry and converts itto a digital signal, which is fed to each of the four sub-channel paths,844, 845, 846, and 847.

[0076]FIG. 8c illustrates the details of sub-channel A, 844 of FIG. 8b.Note that these teachings apply equally to sub-channels B, C, and D,845, 846, and 847, respectively. In complex multiplier 850, thefrequency offset corresponding to the particular sub-channel (+/−2.4 kHzor +/−7.2 kHz) is removed from the incoming signal. The signal is thenfrequency channelized by the square root Nyquist matched filter 852.Once again, matched filter 852 may be any one of the filters discussedherein.

[0077] The filtered signal is passed to Symbol Synchronization block854, which calculates the proper sampling point where there exists no(or minimal) inter-symbol interference signal. This is accomplished bycalculating the magnitude of the sample points over time and selectingthe highest energy points (corresponding to the synchronized symbolsample points). Magnitude tracking (block 858) is performed in order toremove channel effects from the differential decoder by determining if adetected change in amplitude of the signal is based on the intendedsignal information or on fading of the signal caused by interference.Based upon this determination, the threshold by which an incoming pulseis considered to be on the outer or inner ring of the QAM constellation(logically a “1” or a “0”) is modified to adapt to the changing incomingsignal quality.

[0078] Based upon the information provided by magnitude tracker 858, thelikelihood that a bit is in error is calculated in fade finder block859. Blocks determined to be a high risk of being in error are marked as“at-risk” bits in block 861. The “at-risk” bit information is fedforward to the appropriate one of red or blue signal decoding blocks 866and 869 and is used by the decoding blocks' error correcting processes.Symbol synch block 854 also feeds phase tracker 860, which is discussedbelow.

[0079] The modulated signal is fed from magnitude tracker 858 to QAMmodulator/demodulator 864, wherein the signal is de-modulated to thebase band signal before being passed to the appropriate one of red orblue signal decoding paths. Note that only one functional block is shownin FIG. 8c for each element of the path from Nyquist filter 852 to thered/blue decoding blocks. In practice, however two duplicate pathsexist, one each for the red and blue signals. As a consequence Red/Bluemultiplexer 863 is provided in the feedback path between AutomaticFrequency Control block 862 and complex multiplier 850. This is becausea different frequency correction factor will be determined for theincoming red and blue signals. The appropriate correction factor must befed back to the complex multiplier when the desired signal (red or blue)is being received. Frequency control is provided in phase tracker 860and AFC block 862.

[0080] Details of the terminal unit are provided in FIGS. 9a and 9 b. Asnoted above, the following discussion provides additional details and isrelevant to the description of the base unit as both units use similarschemes for voice coding, signal processing, and modulation. FIG. 9aillustrates in block diagram form the terminal unit acting as atransmitter. The end users audio input is received at microphone 902 andpassed to vocoder 904 via codec 903. Vocoder 904 provides for coding,compression, and forward error correction functions, as discussed abovewith reference to FIG. 8b. The signal is then passed to TDM formatter908 along with control and synchronization bits from block 906 as well.The combined signals from vocoder 904 and control and synch block 906are up-converted in TDM formatter block 908 to double the data rate. Thesignal is then passed to channel coder 910, where control and syncinformation is added to the signal and the bits are interleaved in orderto make the transmitted signal less susceptible to noise, as is wellknown in the art.

[0081] The signal is modulated using QAM modulation (as described above)in block 912, as described above with reference to FIGS. 8a through 8 c

[0082] The in-phase and quadrature-phase components of the QAM modulatedsignal are then passed to Nyquist filter 914 which provides a pulseshaping filter in order to limit the overall bandwidth of thetransmitted signal. In the preferred embodiment terminal unit, theNyquist filter operates at a 65 times over-sampling rate, in order tosimplify the analog filtering of the digital image. The Nyquist filter914 can be any of the filters discussed herein and will be matched withthe corresponding filter from the base unit.

[0083] After passing through the Nyquist filter 914, the signal ismultiplied by the frequency offset required for the sub-channel uponwhich the signal is to be transmitted. The offset signal is supplied bysub-channel offset block 926, which selects the subchannel offset basedupon instructions received from the base unit or upon preprogrammedinstructions contained within the terminal unit's non-volatile memory.

[0084] Note that only a single QAM modulator is required for theterminal unit. This is because the terminal unit will only transmit onone sub-channel at any give time, as opposed to the base unit, whichbroadcasts over all sub-channels simultaneously.

[0085] The in-phase signal is fed to D/A converter 916 and thequadrature component is fed to D/A converter 918 where the signals areconverted to analog signals. Filters 920 and 922 filter out spectralimages at the Nyquist rate. Finally, the signals are fed to I & Qmodulator 924 where the signals are modulated to radio frequency beforebeing passed to RF transmitting circuitry (not shown).

[0086]FIG. 9b illustrates the terminal unit functioning as a receiver.Signals from the base unit or another terminal unit are received by RFreceiving circuitry 930 where the RF signal is down-converted andfiltered before being passed to A/D converter and mixer 933 for thein-phase component and 934 for the quadrature-phase component. Also atthis point, the frequency offset associated with sub-channel selectionis removed from the signal components by mixing into the received signala signal complementary to the offset signal. The complementary offsetsignal is determined by sub-channel frequency offset controlinformation, as illustrated by block 926 and depends on the sub-channelupon which the terminal unit is receiving. The digital signals arepassed to a matched square root Nyquist Filter (which comprises any ofthe embodiments described herein) and then demodulated to a real binarysignal in demodulator 932. The digital binary signal is then de-coded inblocks 944 and 946 using Viterbi decoding. The signal is thende-multiplexed in the time domain, wherein the data rate is reduced from16 kb/s to 8 kb/s before being converted to an analog audio signal andreproduced by a speaker or similar end-user interface (not shown), or inthe case of data before being displayed on an end-user interface such asan LCD display. Note that only one time slot 950 or 952 will be activeat any given time and will drive the end-user interface(s).

[0087] Continuous fine frequency control is also provided for asindicated by blocks 860 and 862 of FIG. 8c and block 940 of FIG. 9b.Slot and symbol synchronization is provided for in block 938.

[0088] While this invention has been described with reference toillustrative embodiments, this description is not intended to beconstrued in a limiting sense. Various modifications and combinations ofthe illustrative embodiments, as well as other embodiments of theinvention, will be apparent to persons skilled in the art upon referenceto the description. It is therefore intended that the appended claimsencompass any such modifications or embodiments.

What is claimed is:
 1. A method of generating digital data fortransmission in a communications system, the method comprising:determining a digital value for a signal to be transmitted; determiningan amplitude for each sample in a series of samples by combining thedigital value with a truncated impulse response, the truncated impulseresponse corresponding to the square root of a frequency domain responsewherein the frequency domain response meets the Nyquist criteria andwherein the square root of the frequency domain response has a firstderivative that is continuous over all points; and generating a transmitsignal with a time-varying amplitude based on the series of samples, thetransmit signal having a non-infinite time duration.
 2. The method ofclaim 1 wherein the frequency domain response includes a composite sinefunction.
 3. The method of claim 2 wherein the frequency domainresponse, NF(ω), is represented by the following equations:$\begin{matrix}{{{{NF}(\omega)} = T},{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}} \\{{{{{NF}(\omega)} = {\frac{T}{2}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}} \right\}}} \right)}},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} \leq {\omega } \leq {\frac{\pi}{T}\left( {1 + \alpha} \right)}}}\quad} \\{{{{NF}(\omega)} = 0},{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\end{matrix}$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 4. The method of claim 2 wherein the frequency domainresponse (NF) is represented by the following equations:${N(\omega)} = \left\{ \begin{matrix}{T,{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}} \\{{\frac{T}{2}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{\pi}{2}{\sin \left( {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\rbrack}} \right\}}} \right)},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} < {\omega } < {\frac{\pi}{T}\left( {1 + \alpha} \right)}}} \\{0,{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\end{matrix} \right.$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 5. The method of claim 1 wherein the frequency domainresponse is a Nyquist compliant function that when differentiated can bewritten in the form${F^{\prime}(\omega)} = {{f^{\prime}(\omega)}\cos \left\{ {\frac{\pi}{2}{\sin \left( {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\} {\cos\left\lbrack {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}}$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 6. The method of claim 1 wherein the frequency domainresponse includes a function selected from the group consisting ofhyperbolic sines, hyperbolic cosines and polynomials.
 7. The method ofclaim 1 wherein the frequency domain response is derived by numericalapproximations.
 8. The method of claim 1 wherein the square root of thefrequency domain response has an infinite number of higher orderderivatives, each of the infinite number of higher order derivativesbeing continuous over all points.
 9. An improved Nyquist filter for useas a matched filter in a digital communications system, the filter beingcharacterized in that the frequency domain response meets the Nyquistcriteria and that the square root of the frequency domain response has afirst derivative that is continuous at all points.
 10. The filter ofclaim 9 wherein the frequency domain response is a Nyquist compliantfunction that when differentiated can be written in the form${F^{\prime}(\omega)} = {{f^{\prime}(\omega)}\cos \left\{ {\frac{\pi}{2}{\sin \left( {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\} {\cos\left\lbrack {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}}$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 11. The filter of claim 9 wherein the frequency domainresponse, NF(ω), is represented by the following equations:${{{NF}(\omega)} = T},{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}$${{{NF}(\omega)} = {\frac{T}{2}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}} \right\}}} \right)}},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} \leq {\omega } \leq {\frac{\pi}{T}\left( {1 + \alpha} \right)}}$${{{NF}(\omega)} = 0},{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 12. The filter of claim 9 wherein the frequency domainresponse, NF(ω), is represented by the following equations:${N(\omega)} = \left\{ \begin{matrix}{T,{{{when}\quad {\omega }} \leq {\frac{\pi}{T}\left( {1 - \alpha} \right)}}} \\{{\frac{T}{2}\left( {1 - {\sin \left\{ {\frac{\pi}{2}{\sin \left\lbrack {\frac{\pi}{2}{\sin \left( {\frac{T}{2\quad \alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\rbrack}} \right\}}} \right)},{{{when}\quad \frac{\pi}{T}\left( {1 - \alpha} \right)} < {\omega } < {\frac{\pi}{T}\left( {1 + \alpha} \right)}}} \\{0,{{{when}\quad \frac{\pi}{T}\left( {1 + \alpha} \right)} \leq {\omega }}}\end{matrix} \right.$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 13. A communication device comprising: a digital signalsource; a pulse shaping filter coupled to receive digital data from theanalog-to-digital converter, the pulse shaping filter beingcharacterized in that the frequency domain response meets the Nyquistcriteria and that the square root of the frequency domain response has afirst derivative that is continuous at all points, the pulse shapingfilter having an impulse response corresponding to the square root ofthe frequency domain response; and a modulator coupled to receive asignal from the pulse shaping filter.
 14. The device of claim 13 andfurther comprising a mapping unit coupled between the digital datasource and the pulse shaping filter.
 15. The device of claim 14 whereinthe mapping unit comprises a quadrature amplitude modulation unit andgenerates an I stream of data and a Q stream of data, the I stream ofdata being input to the pulse shaping filter and the Q stream of databeing input to a second pulse shaping filter.
 16. The device of claim 14wherein the mapping unit and the pulse shaping filter are integrated ona single chip.
 17. The device of claim 13 wherein the digital datasource comprises an analog-to-digital converter.
 18. The device of claim13 wherein the pulse shaping filter is implemented with a look-up tablestored in a memory array.
 19. The device of claim 13 and furthercomprising a digital-to-analog converter coupled between the pulseshaping filter and the modulator.
 20. The device of claim 13 wherein thefrequency domain response is a Nyquist compliant function that whendifferentiated can be written in the form${F^{\prime}(\omega)} = {{f^{\prime}(\omega)}\cos \left\{ {\frac{\pi}{2}{\sin \left( {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\} {\cos\left\lbrack {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}}$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 21. A memory device storing a look-up table for animpulse response for a filter, the filter being characterized in thatthe frequency domain response meets the Nyquist criteria and that thesquare root of the frequency domain response has a first derivative thatis continuous at all points.
 22. The device of claim 21 wherein thefrequency domain response is a Nyquist compliant function that whendifferentiated can be written in the form${F^{\prime}(\omega)} = {{f^{\prime}(\omega)}\cos \left\{ {\frac{\pi}{2}{\sin \left( {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right)}} \right\} {\cos\left\lbrack {\frac{T}{2\alpha}\left( {{\omega } - \frac{\pi}{T}} \right)} \right\rbrack}}$

wherein ω is frequency, T is a time period between symbols, and α is aroll-off factor.
 23. The device of claim 21 wherein the memory device isintegrated on the same integrated circuit as a digital signal processorcore.